Assumption REF  is required by [1]} to ensure that the iterates generated by randomized solvers converge in expectation.
It can be easily verified for some sketching matrices.
For example, in randomized Kaczmarz method, \(S = _i\)  with equal probability where \(_i\)  is the \(i\) -th canonical basis of \(\mathbb {R}^{d+m}\) .
Then, we have
\(\mathbb {E}[K_tS(S^TK_t^2S)^\dagger S^TK_t \mid {x}_t,{\lambda }_t] \succeq \frac{\mathbb {E}[K_tS S^TK_t \mid {x}_t,{\lambda }_t]}{\max _j [K_t^2]_{j,j}} = \frac{K_t^2}{(d+m)\cdot \max _j [K_t^2]_{j,j}} \succeq \frac{I}{(d+m)\kappa (K_t^2)},\) 
